Planar lattices do not recover from forest fires
Ioan ManolescuJoint work with Demeter Kiss and Vladas Sidoravicius
8 May 2015
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 1 / 8
Introduction Forest fires
What are forest fires?
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 2 / 8
Introduction Forest fires
What are forest fires?
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 2 / 8
Introduction Forest fires
What are forest fires?
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 2 / 8
Introduction Forest fires
What are forest fires?
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 2 / 8
Introduction Forest fires
What are forest fires?
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 2 / 8
Introduction Forest fires
What are forest fires?
∞Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 2 / 8
Introduction Forest fires
What are forest fires?
∞Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 2 / 8
Introduction Forest fires
What are forest fires?
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . .
and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 2 / 8
Introduction Forest fires
What are forest fires?
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . .
and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 2 / 8
Introduction Forest fires
What are forest fires?
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 2 / 8
Introduction Forest fires
What are forest fires?
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 2 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
Question: is there an infinite connected component?0 1
Supercriticality:Existence of infinite cluster
Subcriticality:No infinite cluster
pc
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
Question: is there an infinite connected component?0 1
Supercriticality:Existence of infinite cluster
Subcriticality:No infinite cluster
Exponential tailfor cluster size
Unique infinite clusterExponential tail for dis-tance to infinite cluster.Trivial large
scale behaviour! Trivial largescale behaviour!
pc
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
Question: is there an infinite connected component?0 1
Supercriticality:Existence of infinite cluster
Subcriticality:No infinite cluster
Exponential tailfor cluster size
Unique infinite clusterExponential tail for dis-tance to infinite cluster.Trivial large
scale behaviour! Trivial largescale behaviour!
CriticalityNo infinite cluster
Scale invariance
pc
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
At pc. . . Crossing probabilities do not degenerate. (RSW)
Existence of critical exponents (arm exponents)
Ppc2n
n[ ]≥ ε∀n,
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
At pc. . . Crossing probabilities do not degenerate. (RSW)
Existence of critical exponents (arm exponents)
Ppc2n
n[ ]≥ ε∀n,
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
At pc. . . Crossing probabilities do not degenerate. (RSW)
Existence of critical exponents (arm exponents)
Ppc2n
n[ ]≥ ε∀n,
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
At pc. . . Crossing probabilities do not degenerate. (RSW)
Existence of critical exponents (arm exponents)
Ppc2n
n[ ]≥ ε∀n,
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
At pc. . . Crossing probabilities do not degenerate. (RSW)
Existence of critical exponents (arm exponents)
Ppc2n
n[ ]≥ ε∀n,
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
At pc. . . Crossing probabilities do not degenerate. (RSW)
Existence of critical exponents (arm exponents)
Ppc2n
n[ ]≥ ε Ppc2n
n[ ]≥ ε∀n,
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction Percolation: a few basic facts
Percolation on Z2 with parameter p ∈ [0, 1]: Pp
vertices are open with probability p, closed with probability 1− p, independently.
At pc. . . Crossing probabilities do not degenerate. (RSW)Existence of critical exponents (arm exponents)
Ppc2n
n[ ]≥ ε Ppc2n
n[ ]≥ ε∀n,
Ppc
n
[ ]≤ n−α1 Ppc
n
[ ]≤ n−(2+λ)
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 3 / 8
Introduction The model
What is self-destructive percolation?
A planar lattice: here Z2.Let p, δ ∈ [0, 1].Two (site) percolation configurations:
ω - intensity p (measure Pp).
σ - intensity δ (small).
ωclose ∞-cluster−−−−−−−−−→ ω
enhancement−−−−−−−→ ωδ = ω ∨ σ.
δc(p) = sup{δ : Pp,δ(0ωδ
←→∞) = 0}.
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 4 / 8
Introduction The model
What is self-destructive percolation?
A planar lattice: here Z2.Let p, δ ∈ [0, 1].Two (site) percolation configurations:
ω - intensity p (measure Pp).
σ - intensity δ (small).
ωclose ∞-cluster−−−−−−−−−→ ω
enhancement−−−−−−−→ ωδ = ω ∨ σ.
δc(p) = sup{δ : Pp,δ(0ωδ
←→∞) = 0}.
∞
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 4 / 8
Introduction The model
What is self-destructive percolation?
A planar lattice: here Z2.Let p, δ ∈ [0, 1].Two (site) percolation configurations:
ω - intensity p (measure Pp).
σ - intensity δ (small).
ωclose ∞-cluster−−−−−−−−−→ ω
enhancement−−−−−−−→ ωδ = ω ∨ σ.
δc(p) = sup{δ : Pp,δ(0ωδ
←→∞) = 0}.
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 4 / 8
Introduction The model
What is self-destructive percolation?
A planar lattice: here Z2.Let p, δ ∈ [0, 1].Two (site) percolation configurations:
ω - intensity p (measure Pp).σ - intensity δ (small).
ωclose ∞-cluster−−−−−−−−−→ ω
enhancement−−−−−−−→ ωδ = ω ∨ σ.
δc(p) = sup{δ : Pp,δ(0ωδ
←→∞) = 0}.
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 4 / 8
Introduction The model
What is self-destructive percolation?
A planar lattice: here Z2.Let p, δ ∈ [0, 1].Two (site) percolation configurations:
ω - intensity p (measure Pp).σ - intensity δ (small).
ωclose ∞-cluster−−−−−−−−−→ ω
enhancement−−−−−−−→ ωδ = ω ∨ σ.
δc(p) = sup{δ : Pp,δ(0ωδ
←→∞) = 0}.
∞
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 4 / 8
Introduction The model
What is self-destructive percolation?
A planar lattice: here Z2.Let p, δ ∈ [0, 1].Two (site) percolation configurations:
ω - intensity p (measure Pp).σ - intensity δ (small).
ωclose ∞-cluster−−−−−−−−−→ ω
enhancement−−−−−−−→ ωδ = ω ∨ σ.
δc(p) = sup{δ : Pp,δ(0ωδ
←→∞) = 0}.
∞
Question: δc(p)→ 0 as p ↘ pc?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 4 / 8
Introduction The model
What is self-destructive percolation?
A planar lattice: here Z2.Let p, δ ∈ [0, 1].Two (site) percolation configurations:
ω - intensity p (measure Pp).σ - intensity δ (small).
ωclose ∞-cluster−−−−−−−−−→ ω
enhancement−−−−−−−→ ωδ = ω ∨ σ.
δc(p) = sup{δ : Pp,δ(0ωδ
←→∞) = 0}.
∞
Theorem [Kiss, M., Sidoravicius] : There exists δ > 0 such that, for all p > pc ,
Pp,δ(infinite cluster in ωδ) = 0.
In particular limp→pc δc(p) > 0
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 4 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
∞Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
∞Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . .
and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . .
and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
t0 tc
no infiniteforest
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
t0 tc
no infiniteforest
infinite forestsappear and burn
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
t0 tc
no infiniteforest
infinite forestsappear and burn
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
t0 tc
no infiniteforest
infinite forestsappear and burn
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
t0 tc
no infiniteforest
infinite forestsappear and burn
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
t0 tc
no infiniteforest
infinite forestsappear and burn
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
Introduction The model
Back to forest fires: non-existence
Trees grow at i.i.d exponential times, instantly - on vacant sites;
As soon as an infinite forest (i.e. cluster) is formed, it instantly burns
Trees then continue to grow . . . and burn
Question: Does this make sense?
t0 tc
no infiniteforest
infinite forestsappear and burn
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 5 / 8
A crossing probability estimate
ω containing crossing
delete crossing cluster−−−−−−−−−−−−→ ω̃enhancement−−−−−−−→ ω̃δ
Proposition
For δ > 0 small enough, as n→∞,
Ppc,δ ( )4n nn
nω ω̃δand → 0
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 6 / 8
A crossing probability estimate
ω containing crossingdelete crossing cluster−−−−−−−−−−−−→ ω̃
enhancement−−−−−−−→ ω̃δ
Proposition
For δ > 0 small enough, as n→∞,
Ppc,δ ( )4n nn
nω ω̃δand → 0
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 6 / 8
A crossing probability estimate
ω containing crossingdelete crossing cluster−−−−−−−−−−−−→ ω̃
enhancement−−−−−−−→ ω̃δ
Proposition
For δ > 0 small enough, as n→∞,
Ppc,δ ( )4n nn
nω ω̃δand → 0
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 6 / 8
A crossing probability estimate
ω containing crossingdelete crossing cluster−−−−−−−−−−−−→ ω̃
enhancement−−−−−−−→ ω̃δ
Proposition
For δ > 0 small enough, as n→∞,
Ppc,δ ( )4n nn
nω ω̃δand → 0
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 6 / 8
A crossing probability estimate
ω containing crossingdelete crossing cluster−−−−−−−−−−−−→ ω̃
enhancement−−−−−−−→ ω̃δ
Proposition
For δ > 0 small enough, as n→∞,
Ppc,δ ( )4n nn
nω ω̃δand → 0
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 6 / 8
Proof of crossing estimate ⇒ Theorem
For δ
′
> 0 small, Ppc,δ ( )4n nn
nω ω̃δand → 0
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 7 / 8
Proof of crossing estimate ⇒ Theorem
For δ
′
> 0 small, Ppc,δ ( )4n nn
nω ω̃δand → 0
For p1 ≥ p2 and δ1, δ2 such that p1 + (1− p1)δ1 ≤ p2 + (1− p2)δ2,
Pp1,δ1 ≤st Pp2,δ2 .
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 7 / 8
Proof of crossing estimate ⇒ Theorem
For δ′> 0 small, ( )4n nn
nω ω̃δand → 0Pp,δ′ , for p ≥ pc .
For p1 ≥ p2 and δ1, δ2 such that p1 + (1− p1)δ1 ≤ p2 + (1− p2)δ2,
Pp1,δ1 ≤st Pp2,δ2 .
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 7 / 8
Proof of crossing estimate ⇒ Theorem
For δ′> 0 small, ( )4n nn
nω ω̃δand → 0Pp,δ′ , for p ≥ pc .
0
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 7 / 8
Proof of crossing estimate ⇒ Theorem
For δ′> 0 small, ( )4n nn
nω ω̃δand → 0Pp,δ′ , for p ≥ pc .
0∞
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 7 / 8
Proof of crossing estimate ⇒ Theorem
For δ′> 0 small, ( )4n nn
nω ω̃δand → 0Pp,δ′ , for p ≥ pc .
0
∞
In both ω̃δ and ωδ!
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 7 / 8
Proof of crossing probability estimate
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
γ - vertical crossing with minimal number of enhanced points.
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
γ - vertical crossing with minimal number of enhanced points.
X = {enhanced points used by γ}. If no crossing X = ∅.
Ppc ,δ(vertical crossing in ω̃δ) =∑
X 6=∅Ppc ,δ(X = X ).
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
Annulus surrounding passage points but not containing passage points:6 arms or 4 half-plane arms in ω (possibly with one defect).
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
Annulus surrounding passage points but not containing passage points:6 arms or 4 half-plane arms in ω (possibly with one defect).
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
Annulus surrounding passage points but not containing passage points:6 arms or 4 half-plane arms in ω (possibly with one defect).
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
Annulus surrounding passage points but not containing passage points:6 arms or 4 half-plane arms in ω (possibly with one defect).
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
Annulus surrounding passage points but not containing passage points:6 arms or 4 half-plane arms in ω (possibly with one defect).
Ppc ( )R
≤(rR
)2+λr Ppc ( )R ≤
(rR
)2+λr
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
For a set X , what is Ppc ,δ(X = X )?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
For a set X , what is Ppc ,δ(X = X )?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
For a set X , what is Ppc ,δ(X = X )?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
For a set X , what is Ppc ,δ(X = X )?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
For a set X , what is Ppc ,δ(X = X )?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
d1 d2
d3
d4
d5
d6
For a set X , what is Ppc ,δ(X = X )?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
For a set X , what is Ppc ,δ(X = X )?
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
d1 d2
d3
d4
d5
d6
Pp,δ(X = X ) ≤ ckn−2−λ∏
j
d−2−λj × δk ,
where d1, . . . , dk are the merger times of X .
#{X with merger times d1, . . . , dk} ≤ C kn2∏
j
dj .
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
P(vetical crossing in ω̃δ) ≤ n−λ∑
X
Pp,δ(X = X )
≤ n−λ∑
k≥1d1,...,dk
(δkck
∏
k
d−1−λk
)= n−λ
∑
k≥1
δc
∑
d≥1
d−1−λ
k
→ 0,
for δ > 0 small.
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8
Proof of crossing probability estimate
P(vetical crossing in ω̃δ) ≤ n−λ∑
X
Pp,δ(X = X )
≤ n−λ∑
k≥1d1,...,dk
(δkck
∏
k
d−1−λk
)= n−λ
∑
k≥1
δc
∑
d≥1
d−1−λ
k
→ 0,
for δ > 0 small.
Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 8 May 2015 8 / 8